Solutions by Kristen Bethke March 2, 2006 16.423J Bone Homework Solution Set PROBLEM 1. Section modulus of idealized femoral shafts: 4 We find the section modulus, Z, by the equation Z = I/r o , where I = 0.25 π (r o 4 – r i ), simplified as 4 Z = (0.25 π r o 4 - 0.25 π ir i )/ r o (1) For adult males of age 25, Z is 2.90 cm 3 . For adult females of age 25, Z is 1.79 cm 3 . Outer radius, r o , as a function of changing inner radius, r i : As years pass, and a human’s osteoblasts and osteoclasts are constantly reforming and absorbing bone. This dynamic process can result in changes in both inner and outer dimensions of the cortical layer of the bone, but bone strength can nevertheless be maintained. If the load on the skeleton remains constant, then the bones can maintain constant strain rate if they maintain a constant section modulus. Consequently, if the inner radius of the cortical layer increases due to bone remodeling, a correspondent increase in outer radius due to bone modeling can prevent a change in bone section modulus. We can see this section modulus maintenance by returning to Equation (1): 4 Z = (0.25 π r o 4 - 0.25 π ir i )/ r o We set Z to be a constant and allow the inner radius r i to increase linearly with time. For this problem r i increases by 0.004 cm/year, a value which we can call r rate . We can collect outer radius, ro, terms on one side and ri terms on the other side of the equation. Then we can solve iteratively for the outer radius ro, which is found implicitly in Equation (2). The variable t is the time in years since the starting conditions. 0.25 π r o 4 – Zr o = 0.25 π (r i,initial + 0.004t) 4 (2) I used an iterative loop in Matlab to calculate the outer radius for each year between age 25 and age 95, subject to the inner radius increasing by 0.004 cm/year and the section modulus remaining constant (at either 2.90 cm 3 for males or 1.79 cm 3 for females). The Matlab code is attached. The results show that it is indeed physically possible for the outer radius to increase in such a way that the section modulus remains the same when the inner radius increases. For a female aging from 25 years old to 80 years old, the outer radius increases from 1.40 cm to 1.49 cm to compensate for an inner radius change from 0.90 cm to 1.12 cm. In other words, in 55 years, a 6.4% increase in r o occurs for a 24% increase in r i . For males going from age 25 to age 80, a 6.5% increase in r o occurs for an 18% increase in r i . Even at age 80, for both females and males, the inner radius is sufficiently smaller than the outer radius as to make physical sense for the bone’s cortical thickness: for females, the thickness (r o – r i ) decreases from 0.50 cm at age 25 to 0.37 cm at age 80.

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Solutions by Kristen Bethke March 2, 2006 16.423J The following figures show the change in cortical layer inner and outer radii with time. Figure 1: Inner and outer radius changes with time for average female, under constant skeletal loading

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